Write a simplified quartic equation with integer coefficients and positive leading coefficient as small as possible, whose single roots are -1/3 and 0 and house a double root at 0.4?

1 Answer
Oct 28, 2017

f(x) = 75x^4-35x^3-8x^2+4x

Explanation:

Let f(x) be our quartic polynomial

We are told that f(x) has roots {-1/3, 0, +0.4, +0.4}

We are also told that f(x) has integer coefficients with the leading coefficient >0

Given the roots, f(x) will have factors of the form: (x+1/3), x, (x-2/5)^2

Given that the coefficients are integer with the coefficient of x^4 >0

f(x) = (3x+1)x(5x-2)^2

= x(3x+1)(25x^2-20x+4)

=x(75x^3 -35x^2-8x+4)

=75x^4-35x^3-8x^2+4x

Since {75, 35, 8, 4} has no common factor greater than 1, f(x) is in its simplest form.